Sandwich structures have the capability to sustain loads beyond the buckling bifurcation point. In light of this, a comprehensive geometrically nonlinear theoretical framework is developed to analyze the structural performance of sandwich plates and shells in this post-buckling regime. The analysis incorporates several structural and material assumptions. The face sheets are modeled as symmetric with respect to both their local and global mid-surfaces and are composed of anisotropic laminated composites. These facings are assumed to be thin and imperfect, allowing the neglect of transverse shear stresses. The core is modeled as a weak-type construction. Within this theoretical context, two types of laminated facings are considered: cross-ply and angle-ply configurations. Two formulations are employed—the mixed formulation and the displacement formulation to capture the complex behavior of the structures. To obtain post-buckling solutions, both the extended Galerkin method and Newton’s method are utilized. Finally, various kinematic and physical parameters are examined with respect to the load-carrying capacity of the sandwich panels. The results provide meaningful insights into the post-buckling behavior and structural performance of these systems.